where \(c_{k\tau \sigma }^{\dagger }\)  creates an electron in the Au substrate
with wave vector \(k\) , pseudospin \(\tau \)  (representing a channel with
symmetry \(3z^{2}-r^{2}\)  for \(\tau =1\)  and \(\pi \)  for \(\tau =-1\) ) and spin \(\sigma \) .
The first term describes the substrate bands, the second the Kondo exchange with the localized spin \(\vec{S}\)  with exchange
couplings \(J_{1}>J_{-1}\) , the third term is the single-ion
uniaxial magnetic anisotropy,
and the last term is the effect of an applied magnetic field \(B\) .
This model is equivalent to that used by Hiraoka et al. [1]}
for large Hund's coupling \(J_{H}\) . As we show below, this model with three
parameters explains all the experimental findings in FePc [2]}. Its fundamental ingredients are well
justified by ab-initio and/or ligand field multiplet calculations: the existence of an almost integer
electronic occupancy of the Fe ion [1]}, [4]}, the spin \(S=1\)  of the FePc molecule [5]}, [1]}, [7]}, [8]}, the markedly
different hybridizations of the \(d_{z^2}\)  and \(d_\pi \)  orbitals with the Au substrate [5]}, [1]}, [11]}, and the
presence of a non-negligible easy-plane magnetic anisotropy. We also
consider that the gold conduction bands have a constant density of states, as first-principle
calculations [12]} do not find any signature of sharp structures (with widths of the order of \(T_K^{(1)}\)  or less),
that would modify sensibly the low energy Kondo physics, and we take the half band width \(W=1\)  as the unit of energy.
